Strategic Network Formation with Attack and Immunization

Abstract

Strategic network formation arises in settings where agents receive some benefit from their connectedness to other agents, but also incur costs for forming these links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization or protection against the attack. An agent's network benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework can be viewed as a stylized model of settings where reachability rather than centrality is the primary interest (as in many technological networks such as the Internet), and vertices may be vulnerable to attacks (such as viruses), but may also reduce risk via potentially costly measures (such as an anti-virus software).The reachability network benefit model has been studied in the setting without attack or immunization [4], where it is known that the set of equilibrium networks is the empty graph as well as any tree. We show that the introduction of attack and immunization changes the game in dramatic ways; in particular, many new equilibrium topologies emerge, some more sparse and some more dense than trees. Our interests include the characterization of equilibrium graphs, and the social welfare costs of attack and immunization.Our main theoretical contributions include a strong bound on the edge density at equilibrium. In particular, we show that under a very mild assumption on the adversary's attack model, every equilibrium network contains at most only 2n − 4 edges for n ≥ 4, where n denotes the number of agents and this upper bound is tight. This demonstrates that despite permitting topologies denser than trees, the amount of "over-building" introduced by attack and immunization is sharply limited. We also show that social welfare does not significantly erode: every non-trivial equilibrium in our model with respect to several adversarial attack models asymptotically has social welfare at least as that of any equilibrium in the original attack-free model.We complement our sharp theoretical results with simulations demonstrating fast convergence of a bounded rationality dynamic, swapstable best response, which generalizes linkstable best response but is considerably more powerful in our model. The simulations further elucidate the wide variety of asymmetric equilibria possible and demonstrate topological consequences of the dynamics, including heavy-tailed degree distributions arising from immunization. Finally, we report on a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium. * The short version of this paper [12] appears in the proceedings of WINE-16.Definition 2. The random attack adversary attacks a vulnerable vertex uniformly at random.So every vulnerable vertex is targeted with respect to the random attack adversary and the adversary induces a distribution over targeted regions such that the probability ...

Network Formation under Random Attack and Probabilistic Spread

“…While Goyal et al [8] show robustness of the structural properties of the original reachability game of Bala and Goyal [2] to a variation with attack, deterministic spread and the option of immunization for players, we show robustness in another variant that involves a cascading attack but disallows immunization. However, on the technical front, the tools that we use to prove these robustness results are very different from the analysis of both of these previous games.…”

“…We start by formalizing our model and borrow most of our notation and terminology from Goyal et al [8]. We assume the n vertices of a graph (network) correspond to individual players.…”

“…Second, Goyal et al [8] study a network formation game where players in addition to having the option of purchasing edges can also purchase immunization from the attack. Since we do not study the effect of immunization purchases in our game, our game corresponds to the regime of parameters in their game where the cost of immunization is so high that no vertex would purchase immunization in equilibria.…”

“…Bala and Goyal [2] study a reachability network game without attacks and show a sharp characterization of equilibrium networks: every tree and the empty network can form in equilibria. Goyal et al [8] study a reachability network formation game where an adversary inspects the formed network and then deliberately attacks a single vertex in the network. The attack then spreads deterministically to neighboring vertices according to a known rule, while agents may immunize against the attack for a fixed cost.…”

Swap Equilibria under Link and Vertex Destruction

“…• Goyal et al [15] consider a model where each player, in addition to building edges, can choose to immunize themselves in exchange for a fee. Then, an adversary selects a connected component of non-immunized vertices to destroy; an alternative description is that the adversary picks a vertex, and then, the destruction spreads from there, while immunized vertices act as firewalls.…”